The processor speed, in megahertz (MHz), of Intel processors...

The processor speed, in megahertz (MHz), of Intel processors can be approximated by the following function of time $t$ in years since the start of $1995:^{65}$ $$ P(t)=\left\{\begin{array}{ll} 180 t+200 & \text { if } 0 \leq t \leq 5 \\ 3,000 t-13,900 & \text { if } 5<t \leq 12 \end{array}\right. $$ a. Is $P$ a continuous function of $t$ ? Why? HINT [See Example 4 of Section 10.3.] b. Is $P$ a differentiable function of $t ?$ Compute $\lim _{t \rightarrow 5^{-}} P^{\prime}(t)$ and $\lim _{t \rightarrow 5^{+}} P^{\prime}(t)$ and interpret the results. HINT [See Before we go on... after Example 5.]

The processor speed, in megahertz (MHz), of Intel processors can be approximated by the following function of time $t$ in years since the start of $1995:^{65}$
$$
P(t)=\left\{\begin{array}{ll}
180 t+200 & \text { if } 0 \leq t \leq 5 \\
3,000 t-13,900 & \text { if } 5<t \leq 12
\end{array}\right.
$$
a. Is $P$ a continuous function of $t$ ? Why? HINT [See Example 4 of Section 10.3.]
b. Is $P$ a differentiable function of $t ?$ Compute $\lim _{t \rightarrow 5^{-}} P^{\prime}(t)$ and $\lim _{t \rightarrow 5^{+}} P^{\prime}(t)$ and interpret the results. HINT [See Before we go on... after Example 5.]

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Key Concepts

One-sided Derivatives

One-sided derivatives refer to the limits of the derivative as the independent variable approaches a particular point from one side (left-hand or right-hand). In the context of piecewise functions, calculating these limits helps determine whether the function’s rate of change is consistent across the boundary, which is necessary for differentiability.

Differentiability

Differentiability of a function at a point requires that the derivative exists at that point. For piecewise functions, even if the function is continuous at the boundary, the derivatives from the left and right must also agree. If they differ, the function fails to be differentiable at that boundary.

Continuity

Continuity of a function means that there are no breaks or jumps at any point in its domain. For piecewise functions, this involves ensuring that the limits from the left and right at the boundary between pieces are equal and match the function's value at that point.

Piecewise Functions

Piecewise functions are defined by different expressions over different intervals of the independent variable. They require careful analysis at the boundaries between pieces to determine properties like continuity and differentiability, as these properties might change at the points where the definition of the function switches.

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